In general you can "sandwich" a monad in between two halves of an adjunction.
That is to say, if you have an adjunction F -| G : C -> D then not only does GF
form a monad, but GMF forms a monad for M a monad in D. Therefore if we
have an adjunction F -| G : Hask -> Hask^op then we can lift a Comonad in Hask
which is a Monad in Hask^op to a Monad in Hask.

For any r, the Contravariant functor / presheaf (-> r) :: Hask^op -> Hask is adjoint to the "same"
Contravariant functor (-> r) :: Hask -> Hask^op. So we can sandwhich a
Monad in Hask^op in the middle to obtain w (a -> r-) -> r+, and then take a coend over
r to obtain forall r. w (a -> r) -> r. This gives rise to Co. If we observe that
we didn't care what the choices we made for r were to finish this construction, we can
upgrade to forall r. w (a -> m r) -> m r in a manner similar to how ContT is constructed
yielding CoT.

We could consider unifying the definition of Co and Rift, but
there are many other arguments for which Rift can form a Monad, and this
wouldn't give rise to CoT.